Theorem mobii 1985

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 1984	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1298	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 103  ⊤wtru 1290  ∃*wmo 1949

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-eu 1951  df-mo 1952

This theorem is referenced by:  moaneu  2024  moanmo  2025  2moswapdc  2038  2exeu  2040  rmobiia  2556  rmov  2639  euxfr2dc  2800  rmoan  2815  2rmorex  2821  mosn  3479  dffun9  5044  funopab  5049  funco  5054  funcnv2  5074  funcnv  5075  fun2cnv  5078  fncnv  5080  imadif  5094  fnres  5130  ovi3  5781  oprabex3  5900  frecuzrdgtcl  9819  frecuzrdgfunlem  9826  fisum  10778